4 "Quick" Questions (1 Viewer)

[Doctor Jolly]

There are just some things that I don't get:

1. Find the exact value of 2cos105° sin105°
answer: -1/2

2. Show that sin²7Θ - sin²4Θ = sin11Θsin3Θ

In the answers for this, I understood everything up until the bolded bit:

RHS:
= sin11Θsin3Θ
= sin(7Θ+4Θ)sin(7Θ-4Θ)
= sin²7Θ - sin²4Θ - cos²7Θsin²4Θ
= sin²7Θ(1-sin²4Θ) - (1 - sin²7Θ)sin²4Θ -- how did the take sin²7Θ out of sin²4Θ to make sin²7Θ(1-sin²4Θ)?
= sin²7Θ - sin²7Θsin²4Θ - sin²4Θ + sin²7Θsin²4Θ
= sin²7Θ - sin²4Θ
= LHS

3. Find the number of arrangements of the letters in the word TUESDAY if there are exactly three letters between A and E.
answer: 720

Also, can anyone explain to me when you would use the permutation formula? In all the permutation exercises I've done, I haven't used it, yet I got the answers right ...

4. See attached image:

Sorry if this seems like a lot, but it's just an accumulation of things I didn't get over the past week

Thanks for any help!

 

[wendus]

I FKN HATE PERMUTATIONS & COMBS

btw sorry i can't answer any of your q's atm. but you're hsc is next year, so relax for the time being. it could only get worse, trust me. key with maths is continual and non-stop practise, which i hate.

 

[Studentleader]

1. Find the exact value of 2cos105° sin105°
answer: -1/2

cos105 = -cos(180-105) = -cos(75)=-cos(30+45)
=cos30cos45-sin30sin45
=(sqrt3)/2 * 1/(sqrt2) - 1/2 * 1/(sqrt2)
=(sqrt3/(2(sqrt2)))-1/(2sqrt2)
= ((sqrt3)-1)/(2sqrt2)
sin105=sin(75)=sin(30+45)
=sin30cos45+cos30sin45
=1/2 * 1/(sqrt2) + (sqrt3)/2 * 1/sqrt(2))
=1+(sqrt3)/(2sqrt2)

multiply the bold bits to get -0.25
-0.25x2 = -1/2

3. Find the number of arrangements of the letters in the word TUESDAY if there are exactly three letters between A and E.
answer: 720

2x(3x(5x4x3x2x1))
_____^^^^^^^because n(letters MINUS those needed to be arranged) = 5
__^^3 places where the 1st letter can be placed (A___E???,?A___E?? and ??A___E) where ? is any other letter and ___ is 3 letters
^^2x because A and E are interchangable


Cant help with the others, dont like the multiple angles

 

[Undermyskin]

It's rather late now so I can only quickly check these.

1st: correct. 2 sin105 cos105 = sin210 = sin(-30) = -1/2. remember 2sinAcosA = sin 2A.

2nd: line 3. there's a prob there. You made a mistake when breaking the bracket. Plus your work is weird. You jump in the steps you next to show clearly and show TOO much in the steps you should shorten.

3rd: Correct. Hell, how can you get the answer then? OK. Simple: suppose E is always in front of A. There are 7 spaces, so there are only 3 possible cases for the 2 to have 3 letters between, and also 2 letters outside. Ok, considering the 3 'inside' spaces, there are 5P3 ways to arrange other 5 letters. For the other 2 spaces, there are 2P2 or simply 2! ways. Time the 2 together: 5P3 * 2P2 = 120.

As said, there are 3 possible ways to arrange E...A --> 120 * 3 = 360 ways. However, there's the case when A is in front of E --> 360*2 = 720.

4th: it's sheer formula.

...= tan (7O-3O) = tan4O

 

[lolokay]

2. Show that sin²7Θ - sin²4Θ = sin11Θsin3Θ

In the answers for this, I understood everything up until the bolded bit:

RHS:
= sin11Θsin3Θ
= sin(7Θ+4Θ)sin(7Θ-4Θ)
= sin²7Θ - sin²4Θ - cos²7Θsin²4Θ
= sin²7Θ(1-sin²4Θ) - (1 - sin²7Θ)sin²4Θ -- how did the take sin²7Θ out of sin²4Θ to make sin²7Θ(1-sin²4Θ)?
= sin²7Θ - sin²7Θsin²4Θ - sin²4Θ + sin²7Θsin²4Θ
= sin²7Θ - sin²4Θ
= LHS

sin(7x + 4x)sin(7x - 4x)
= (sin7xcos4x + sin4xcos7x)(sin7xcos4x - sin4xcos7x)
= sin^2 7x cos^2 4x - sin^2 4x cos^2 7x
= sin^2 7x (1 - sin^2 4x) - sin^2 4x (1 - sin^2 7x)
= sin^2 7x - sin^2 4x sin^2 7x - sin^2 4x + sin^2 4x sin^2 7x
= sin^2 7x - sin^2 4x

it's the line before the bolded line that doesn't make sense. The bolded line would have been just converting the cos^2 x's to (1 - sin^2 x)'s

Also, can anyone explain to me when you would use the permutation formula? In all the permutation exercises I've done, I haven't used it, yet I got the answers right ...

what's the permutation formula? is it just n!/(n-k)! ? what do you do to solve them?

 

[pLuvia]

You usually don't use the permutation formula directly or alone in exams, they are usually accompanied by combinations. But effectively when you use the combination formula you most likely use it with a number with a factorial i.e. picking some items from a selection then arranging it like in that question you asked.

As you know the permutation formula is just the combination formula multiplied by the number of arrangements

 

[Mark576]

cos105 = -cos(180-105) = -cos(75)=-cos(30+45)
=cos30cos45-sin30sin45
=(sqrt3)/2 * 1/(sqrt2) - 1/2 * 1/(sqrt2)
=(sqrt3/(2(sqrt2)))-1/(2sqrt2)
= ((sqrt3)-1)/(2sqrt2)
sin105=sin(75)=sin(30+45)
=sin30cos45+cos30sin45
=1/2 * 1/(sqrt2) + (sqrt3)/2 * 1/sqrt(2))
=1+(sqrt3)/(2sqrt2)

multiply the bold bits to get -0.25
-0.25x2 = -1/2

OR alternatively:

2sin(105)cos(105) = sin(210) = sin(180+30) = -1/2

 

[kaz1]

OR alternatively:

2sin(105)cos(105) = sin(210) = sin(180+30) = -1/2

Winner.

 

[Doctor Jolly]

D'oh.

I get it all now.

Thanks everyone!

I can't believe how blind I was ..

 

[Studentleader]

Winner.

I agree, probably best to over elaborate the working though x)